abstract algebra – $ mathbb {Q} ( zeta_ {10}) $ field extension over $ mathbb {Q} $

I want to calculate the degree of $ mathbb {Q} ( zeta_ {10}) $ over $ mathbb {Q} $,

Use the dimension set: $[mathbb{Q}(zeta_{10}) : mathbb{Q}]=[mathbb{Q}(zeta_{10}) : mathbb{Q(zeta_5)}] cdot[mathbb{Q}(zeta_{5}) : mathbb{Q}] = 2 cdot4 = 8 $, Due to the fact that $ x ^ 4 + x ^ 3 + x ^ 2 + x + 1 $ is irreducible and $ x ^ 2- zeta_5 $ is the minimum polynomial for $ zeta_ {10} $ over $ mathbb {Q} ( zeta_5) $,

However: $ x ^ 5 + 1 | _ { zeta_ {10}} = {(e ^ { frac {2 pi i} {10}})} ^ 5 + 1 = e ^ { pi i} + 1 = 0 $,

I am confused because the extension should be $ 8 $but it seems so $ zeta_ {10} $ is a root of $ x ^ 5 + 1 $,